\begin{thebibliography}{9}


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H. Goldstein, C. P. Poole, and J. L. Safko,
\emph{Classical mechanics},
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\bibitem{classical_electromagnetism}
J. D. Jackson,
\emph{Classical electrodynamics},
Wiley (1998).

\bibitem{quantum_mechanics1}
F. Schwabl,
\emph{Quantum Mechanics},
Springer (2007).

\bibitem{schrodinger1}
E. Schr\"odinger,
\emph{Quantisierung als Eigenwertproblem},
Annalen der Physik {\bf{79}}, 361-376 (1926).

\bibitem{born1}
M. Born,
\emph{Zur Quantenmechanik der Sto$\beta$vorg\"ange},
Z. Phys. {\bf{37}}, 863 (1926).

\bibitem{quantum_mechanics}
F. Schwabl,
\emph{Advanced Quantum Mechanics},
Springer (2008).

\bibitem{bose}
S. N. Bose,
\emph{Plancks Gesetz und Lichtquantenhypothese}, 
Zeitschrift f\"ur Physik {\bf{26}}, 178-181 (1924).

\bibitem{fermi}
E. Fermi,
\emph{Sulla quantizzazione del gas perfetto monoatomico},
Rendiconti Lincei {\bf{3}}, 145-9 (1926).

\bibitem{dirac}
P. A. M. Dirac,
\emph{On the theory of quantum mechanics},
Proc. Royal Soc. A {\bf{112}}, 661-77 (1926).

\bibitem{statistical_mechanics}
R. K. Pathria,
\emph{Statistical mechanics},
Elsevier (2001).

\bibitem{bloch0}
F. Bloch,
\emph{\"Uber die Quantenmechanik der Elektronen in Kristallgittern},
Z. Phys. {\bf{52}}, 555 (1928).

\bibitem{physics_of_semiconductor_devices}
S. M. Sze, K. K. Ng,
\emph{Physics of semiconductor devices},
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\bibitem{transistor}
A. G. Levine, 
\emph{John Bardeen, William Shockley, Walter Brattain�Invention of the Transistor},
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\bibitem{how_transistors_work}
N. Chandler,
\emph{How transistors work},
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S. J. Tans, A. R. M. Verschueren, and C. Dekker,
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Nature {\bf{393}}, 49-52 (1998).

\bibitem{biotransistor}
J. Bonnet, P. Yin, M. E. Ortiz, P. Subsoontorn, and D. Endy,
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\bibitem{KS}
W. Kohn, and L. J. Sham,
\emph{Self-consistent equations including exchange and correlation effects},
Physical Review {\bf{140}}, A1133-A1138 (1965).

\bibitem{GW}
L. Hedin, 
\emph{New method for calculating the one-particle Green's function with application to the electron-gas problem}, 
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\bibitem{fulde1}
P. Fulde,
\emph{Electron correlations in molecules and solids},
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\bibitem{fm1_wiki}
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J. G. Bednorz, and K. A. M\"uller,
\emph{Possible high T$_c$ superconductivity in the Ba-La-Cu-O system},
Z. Phys. B {\bf{64}}, 189-193 (1986).

\bibitem{schilling}
A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott,
\emph{Superconductivity above 130 K in the Hg-Ba-Ca-Cu-O system},
Nature {\bf{363}}, 56-58 (1993).

\bibitem{kamihara}
Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, 
\emph{Iron-based layered superconductivity La[O$_\mathrm{1-x}$F$_\mathrm{x}$]FeAs (x=0.05-0.12) with T$_\mathrm{c}$ = 26 K"},
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\bibitem{hubbard}
J. Hubbard, 
\emph{Electron correlations in narrow energy bands},
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Defense Sciences Office,
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\bibitem{yamashita1} 
M. Yamashita, 
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\emph{Quantum optics},
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\bibitem{greiner1}
M. Greiner,
\emph{Ultracold quantum gases in three-dimensional optical lattice potentials}, 
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\bibitem{solid_state}
G. Grosso, and G. P. Parravinci,
\emph{Solid State Physics},
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\bibitem{jaksch1} 
D. Jaksch, 
\emph{Bose-Einstein Condensation and Applications}, 
Ph.D thesis, Universit\"at Innsbruck (1999).

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C. Chin, R. Grimm, P. Julienne, E. Tiesinga,
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\bibitem{trotsky}
S. Trotzky,	 L. Pollet,	F. Gerbier, U. Schnorrberger, I. Bloch, N. V. Prokof�ev, B. Svistunov, and M. Troyer,
\emph{Suppression of the critical temperature for superfluidity near the Mott transition},
Nature Physics {\bf{6}}, 998-1004 (2010) 

\bibitem{greiner2}
M. Greiner, O. Mandel, T. Esslinger, T. W. H\"ansch, and I. Bloch,
\emph{Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms},
Nature {\bf{415}}, 39-44 (2002).

\bibitem{fisher}
M. P. A. Fisher, P. Weichman, G. Grinstein, and D. S. Fisher,
\emph{Boson localization and the superfluid-insulator transition},
Phys. Rev. B {\bf{40}}, 546 (1989).

\bibitem{chin2}
N. Gernelke, X. Zhang, C-L. Hung, and C. Chin,
\emph{In situ observation of incompressible Mott-insulating domains in ultracold atomic gases},
Nature {\bf{460}}, 995 (2009). 

\bibitem{bloch1}
J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr,
\emph{Single-atom-resolved fluorescence imaging of an atomic Mott insulator},
Nature {\bf{467}}, 68 (2010).

\bibitem{shiang1}
S. Fang, C-M. Chung, P. N. Ma, P. Chen, and D-W. Wang,
\emph{Quantum criticality from in situ density imaging}, 
Phys. Rev. A {\bf{83}}, 031605 (2011). 

\bibitem{jaksch1}
D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller,
\emph{Cold bosonic atoms in optical lattices},
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\bibitem{muramatsu1}
R. Staudt, M. Dzierzawa, and A. Muramatsu
\emph{Phase diagram of the three-dimensional Hubbard model at half filling},
Eur. Phys. Jour. B {\bf{17}}, 411-415 (2000).

\bibitem{fermion_ol_experiment}
R. J\"ordens, N. Strohmaier, K. G\"unter, H. Moritz, and T. Esslinger,
\emph{A Mott insulator of fermionic atoms in an optical lattice},
Nature {\bf{455}}, 204-207 (2008).


%%% References cited in and after Chapter 2

\bibitem{prokofev1}
N. V. Prokof'ev, B. V. Svistunov, and I. S. Tupitsyn, 
\emph{Exact, complete, and universal continuous-time worldline Monte Carlo approach to the statistics of discrete quantum systems},
Sov. Phys. JETP {\bf{87}}, 310 (1998).

\bibitem{pollet1}
L. Pollet, K. V. Houcke, and S. M. A. Rombouts
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J. Comp. Phys. {\bf{225/2}}, 2249-2266 (2007).

\bibitem{fetter}
A. L. Fetter, and J. D. Walecka,
\emph{Quantum theory of many-particle systems},
Dover (2003).

\bibitem{markov_chain}
J. R. Norris,
\emph{Markov chains},
Cambridge University Press (1998).

\bibitem{probability_theory}
M. Taboga,
\emph{Lecture on probability theory and mathematical statistics},
Createspace (2012).

\bibitem{metropolis}
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,
\emph{Equation of state calculations by fast computing machines},
Jour. Chem. Phys. {\bf{21}}, 1087 (1953).

%%% References cited in and after Chapter 3

\bibitem{thermometry}
P. N. Ma, L. Pollet, and M. Troyer,
\emph{Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice with in-situ density imaging},
Phys. Rev. A {\bf{82}}, 033627 (2010).

\bibitem{bloch2}
I. Bloch, J. Dalibard, and W. Zwerger,
\emph{Many body physics with ultracold gases},
Rev. Mod. Phys. {\bf{80}}, 885 (2008).

\bibitem{zhou1}
Q. Zhou, and T-L. Ho,
\emph{Universal thermometry foe quantum simulation},
Phys. Rev. Lett. {\bf{106}}, 225301 (2011).

\bibitem{zhou2}
Q. Zhou, Y. Kato, N. Kawashima, and N. Trivedi,
\emph{Direct mapping of the finite temperature phase diagram of strongly correlated quantum models},
Phys. Rev. Lett. {\bf{103}}, 085701 (2009).

%%% References cited in and after Chapter 4

\bibitem{implementing_dwa}
P. N. Ma, L. Pollet, and M. Troyer,
\emph{Fast implemention of the directed worm algorithm for optical lattices},
in preparation (2013).


%%% References cited in and after Chapter 5

\bibitem{dft_fermigas}
P. N. Ma, S. Pilati, M. Troyer, and X. Dai,
\emph{Density functional theory for atomic Fermi gases},
Nature Phys. {\bf{8}}, 601 (2012).

\bibitem{hk_theorem}
P. Hohenberg, and W. Kohn,
\emph{Inhomogeneous electron gas},
Phys. Rev {\bf{136}}, B864 (1964).

\bibitem{ks}
W. Kohn, and L. J. Sham,
\emph{Self-consistent equations including exchange and correlation effects},
Phys. Rev. {\bf{140}}, A1133 (1965).

\bibitem{group_theory}
F. A. Cotton,
\emph{Chemical applications of group theory},
Wiley (1990).

\bibitem{dmft}
A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, 
\emph{Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions},
Rev. Mod. Phys. {\bf{68}}, 13-125 (1996).
 

%%% References cited in and after Chapter 6

\bibitem{fowler}
M. Fowler,
\emph{Historical beginnings of theories of electricity and magnetism},
\url{http://galileo.phys.virginia.edu/classes/109N/more_stuff/E&M_Hist.pdf}, Uni. Virginia (1997).

\bibitem{relativity}
A. Einstein,
\emph{Zur Elektrodynamik bewegter K\"orper},
Ann. Phys. {\bf{322}}, 891-921 (1905).

\bibitem{yosida}
K. Yosida,
\emph{Theory of magnetism},
Springer (1998).

\bibitem{mit}
G. B. Jo, Y. R. Lee, J. H. Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle,
\emph{Itinerant ferromagnetism in a Fermi gas of ultracold atoms},
Science {\bf{325}}, 1521 (2009).


%%% References cited in and after Appendix A

\bibitem{alps_webpage}
P. N. Ma, L. Pollet, and M. Troyer,
\emph{Simulating the boson Hubbard model using QMC Directed-Worm-Algorithm (DWA) code}
The ALPS project, \url{http://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:MC-09_Directed_Worm_Algorithm} (2013).


%%% References cited in and after Appendix B

\bibitem{astra2004}
G.~E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, 
\emph{Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study},
Phys. Rev. Lett. {\bf{93}}, 200404 (2004).

\bibitem{carlson2004}
S.Y.- Chang, V.~R. Pandharipande, J. Carlson, and K.~E. Schmidt, 
\emph{Quantum Monte Carlo studies of superfluid Fermi gases},
Phys. Rev. A {\bf{70}}, 043602 (2004).

\bibitem{carlson2005}
J. Carlson, and S. Reddy,
\emph{Asymmetric two-component fermion systems in strong coupling},
Phys. Rev. Lett {\bf{95}}, 060401 (2005).

\bibitem{lobo}
C. Lobo, A. Recati, S. Giorgini, and S. Stringari,
\emph{Normal state of a polarized Fermi gas at unitarity},
Phys. Rev. Lett {\bf {97}}, 200403 (2006).

\bibitem{pilati2008}
S. Pilati, and S. Giorgini,
\emph{Phase separation in a polarized Fermi gas at zero temperature},
Phys. Rev. Lett. {\bf{100}} (030401) (2008).

\bibitem{navon}
N. Navon, S. Nascimbene, F. Chevy, and C. Salomon, 
\emph{The equation of state of a low-temperature Fermi gas with tunable interactions},
Science {\bf{328}}, 729-732 (2010).

\bibitem{nascimbenepilati}
S. Nascimb\`ene, \emph{et~al.}
\emph{Fermi-liquid behavior of the normal phase of a strongly interacting gas of cold atoms}.
Phys. Rev. Lett. {\bf{106}}, 215303 (2011).

\bibitem{dmc_solids}
W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, 
\emph{Quantum Monte Carlo simulations of solids},
Rev. Mod. Phys. {\bf{73}}, 33-83 (2001).

\bibitem{pilati2010}
S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer,
\emph{Itinerant ferromagnetism of a repulsive atomic Fermi gas: A Quantum Monte Carlo study},
Phys. Rev. Lett. {\bf{105}}, 030405 (2010).

\bibitem{huang}
K. Huang, and C. N. Yang, 
\emph{Quantum-mechanical many-body problem with hard-sphere interaction},
Phys. Rev. {\bf{105}}, 767-775 (1957).

\bibitem{lee}
T. D. Lee, and C. N. Yang, 
\emph{Many-body problem in quantum mechanics and quantum statistical mechanics},
Phys. Rev. {\bf{105}}, 1119-1120 (1957).

\bibitem{duine}
R. A. Duine, and A. H. MacDonald, 
\emph{Itinerant ferromagnetism in an Ultracold atom Fermi gas},
Phys. Rev. Lett. {\bf{95}}, 230403 (2005).
 
\bibitem{recati}
A. Recati, and S. Stringari, 
\emph{Spin fluctuations, susceptibility, and the dipole oscillation of a nearly ferromagnetic fermi gas},
Phys. Rev. Lett. {\bf{106}}, 080402 (2011).

\bibitem{kanno2}
S. Kanno, 
\emph{Criterion for the ferromagnetism of hard sphere fermi liquid},
Prog. Theo. Phys. {\bf{44}}, 813-815 (1970).

\bibitem{mora}
C. Mora, and F. Chevy, 
\emph{Normal phase of an imbalanced Fermi gas},
Phys. Rev. Lett. {\bf{104}}, 230402 (2010).

\bibitem{cui}
X. Cui, and H. Zhai, 
\emph{Stability of a fully magnetized ferromagnetic state in repulsively interacting ultracold Fermi gases},
Phys. Rev. A {\bf{81}}, 041602 (2010).

\bibitem{recatiprivate}
A. Recati,
\emph{Private communication},
(2010).

\bibitem{pilati2012}
S. Pilati, and M. Troyer,
\emph{Bosonic superfluid-insulator transition in continuous space},
Phys. Rev. Lett. {\bf{108}}, 155301 (2012).

\end{thebibliography}
